Characterization of isometric operators through remotality

We obtain the following characterization of Hilbert spaces. Let E be a Banach space whose unit sphere S has a hyperplane of symmetry. Then E is a Hilbert space iff any of the following two conditions is fulfilled: a) the isometry group Iso E of E has a dense orbit in S; b) the identity component G 0 of the group Iso E endowed with the strong operator topology acts topologically irreducible on E. Some related results on infinite dimentional Coxeter groups generated by isometric reflexions are given which allow to analyse the structure of isometry groups containing sufficiently many reflexions.

2004

Let X and Y be Banach spaces, and L(X, Y ) be the spaces of bounded linear operators from X into Y. In this paper we give full characterization of isometric onto operators of L(X, Y ), for a certain class of Banach spaces, that includes p , 1 < p < ∞. We also characterize the isometric onto operators of L(c 0 ) and K( 1 ), the compact operators on 1 . Furthermore, the multiplicative isometric onto operators of L( 1 ), when multiplication on L( 1 ) is taken to be the Schur product, are characterized.

Springer Optimization and Its Applications, 2011

We analyze the problem of stability of linear isometries (SLI) of Banach spaces. Stability means the existence of a function σ (ε) such that σ (ε) → 0 as ε → 0 and for any ε-isometry A of the space X (i.e., (1 − ε) x ≤ Ax ≤ (1 + ε) x for all x ∈ X) there is an isometry T such that A − T ≤ σ (ε). It is known that all finite-dimensional spaces, Hilbert space, the spaces C(K) and L p (µ) possess the SLI property. We construct examples of Banach spaces X, which have an infinitely smooth norm and are arbitrarily close to the Hilbert space, but fail to possess SLI, even for surjective operators. We also show that there are spaces that have SLI only for surjective operators. To obtain this result we find the functions σ (ε) for the spaces l 1 and l ∞. Finally, we observe some relations between the conditional number of operators and their approximation by operators of similarity.

Characterization of isometries in l^2 space, 2022

In general sense, we can define the isometries as transformations which preserve distance between elements. In this paper, we show a characterization of isometries in l^2(X)(for a nonempty set X) which is a Hilbert space .

Bulletin of the Brazilian Mathematical Society, New Series, 2006

Let X be a real Hilbert space with dim X ≥ 2 and let Y be a real normed space which is strictly convex. In this paper, we generalize a theorem of Benz by proving that if a mapping f , from an open convex subset of X into Y , has a contractive distance ρ and an extensive one Nρ (where N ≥ 2 is a fixed integer), then f is an isometry.

IOSR Journals , 2019

In this paper, we study that two Banach spaces are isometrically iso- morphic if Hausdorff distance between them measures zero.

1997

Let $X$ be a separable $L_1$ or a separable $C(K)$-space, and let $Y$ be any Banach space. $I(X, Y )$ denotes the set of all isometries from $X$ to $Y$ . showed that for any finite measure space $(\Omega,\mu)$ and any $1 &lt; p &lt;\infty$, every isometry $T : X\rightarrow L_p(\Omega, Y )$ has the form $$T x(t) = h(t)U(t)x ,$$ where $h\in L_p$ with $\parallel h\parallel_p = 1$ and $U$ is a strongly measurable function from $\Omega$ into $I(X, Y )$. In this article, we extend this result to the Köthe-Bochner function spaces $E(Y )$ when $E$ is strictly convex. We also show that every isometry from $\ell^n_\infty$ into $E(Y $) has the above form if $n\geq 3$ and $E$ is a strictly monotone Köthe function space.

Linear and Multilinear Algebra, 2018

Let X, Y be compact Hausdorff spaces and E, F be Banach spaces over R or C. In this paper, we investigate the general form of surjective (not necessarily linear) isometries T : A −→ B between subspaces A and B of C(X, E) and C(Y, F), respectively. In the case that F is strictly convex, it is shown that there exist a subset Y 0 of Y , a continuous function Φ : Y 0 −→ X onto the set of strong boundary points of A and a family {V y } y∈Y0 of real-linear operators from E to F with V y = 1 such that T f (y) − T 0(y) = V y (f (Φ(y))) (f ∈ A, y ∈ Y 0). In particular, we get some generalizations of the vector-valued Banach-Stone theorem and a generalization of Cambern's result. We also give a similar result in the case that F is not strictly convex, but its unit sphere contains a maximal convex subset which is singleton.

Journal of Mathematical Analysis and Applications, 2004

The aim of this paper is to study the set I r X of isometric reflection vectors of a real Banach space X. We deal with geometry of isometric reflection vectors and parallelogram identity vectors, and we prove that a real Banach space is a Hilbert space if the set of parallelogram identity vectors has nonempty interior. It is also shown that every real Banach space can be decomposed as an I r -sum of a Hilbert space and a Banach space with some points which are not isometric reflection vectors. Finally, we give a new proof of the Becerra-Rodríguez result: a real Banach space X is a Hilbert space if and only if I r X is not rare.